Which of the following formulas could represent a function with a domain of all Real numbers?

y=(1)/(x),  y=x,  y=\sqrt(x),   x=5

To determine which of the given formulas could represent a function with a domain of all real numbers, let's analyze each one:

1. **\( y = \frac{1}{x} \)**:
   - This is a rational function. The function is undefined when \( x = 0 \) because division by zero is undefined. Therefore, the domain is all real numbers **except** \( x = 0 \).
   - **Domain**: \( \mathbb{R} \setminus \{0\} \).

2. **\( y = x \)**:
   - This is a linear function. It is defined for all real values of \( x \).
   - **Domain**: \( \mathbb{R} \) (all real numbers).

3. **\( y = \sqrt{x} \)**:
   - This is a square root function. The square root is only defined for non-negative values of \( x \), meaning \( x \geq 0 \).
   - **Domain**: \( [0, \infty) \) (all non-negative real numbers).

4. **\( x = 5 \)**:
   - This represents a vertical line, which is not a function because for a function, each \( x \) value should map to exactly one \( y \) value. Here, \( x = 5 \) is a constant and does not represent \( y \) as a function of \( x \).

### Conclusion
The only formula that represents a function with a domain of all real numbers is **\( y = x \)**.

Would you like further details on this, or do you have any questions?

Here are 5 related questions to consider:
1. What is the range of the function \( y = x \)?
2. How do we determine the domain and range of a function?
3. What are the characteristics of a linear function?
4. What happens to the graph of \( y = \frac{1}{x} \) near \( x = 0 \)?
5. How can we modify \( y = \sqrt{x} \) to make its domain all real numbers?

**Tip:** When determining the domain of a function, always consider values that might cause the function to be undefined, such as division by zero or taking the square root of a negative number.

Explore which mathematical formulas represent functions with a domain of all real numbers. Learn about rational, linear, and square root functions, and understand why a vertical line does not qualify as a function. Suitable for Grades 9-12.

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